Coin thrown 100 times Find the probability that coat does no

Coin thrown 100 times. Find the probability that \"coat\" does not fall by more than 50 cases. Use the limit of the integral theorem of Laplace.

Solution

There are 101 possibilities for the number of \"coats\": 0 coats, 1 coat,...,100 coats. These are not equally likely, but there are pairs that are equally likely. More specifically, the probability of 0 coats is the same as 100 coats, the probability of 1 coat is the same as 99 coats, etc. These all add to 1. If we let [P_n] be the probability of [n] coats, then we have

[P_0+P_1+...+P_99+P_100]

[=(P_0+P_100)+(P_1+P_99)+...+(P_49+P_51)+P_50]

[=2(P_0+P_1+...+P_49)+P_50=1,]

and we wish to find [P_0+P_1+...+P_50.] We can get [P_50] fairly easily by calculating [_100 C_50 (1/2)^100~~0.080.]

Then we solve for [P_0+...+P_49~~0.92/2=0.46.] Thus the probability of getting no more than 50 coats is approximately

[0.54,] and the exact answer is found by keeping the exact answer for [P_50] instead of using [0.08.]